Can a square number be expressed as sum of squares of two other members.?

Question

Is there any theorem to tell if square of a number can be expressed as sum of squares of two other distinct numbers.

I have one such set. ${5, 4, 3}$

$5^2 = 4^2 + 3^2$

Given a number $n$ how to find if the above conditions satisfy ?

Answer 1

There is the concept of Pythagorean Triples. for:

$$a^2+b^2=c^2$$

you can use the following values for a,b and c where m and n are any two positive integers such that $m>n$:

$$a^2=m^2-n^2 \; and \; b=2mn\; and \; c^2=m^2+n^2 $$

Here is a sample:

You can read more about the subject in wiki-Pythagorean_triple and for a better method to generate the numbers in here: A Direct Method To Generate Pythagorean Triples .

Answer 2

These sets are called pythagorean triplets for obvious reasons. If {x, y, z} are p. triplets so are any {nx, ny, nz}. If m is odd there's a way to construct a triplet that I'm holding off on sharing but I'll tell you two of the terms are consecutive.